For Republicans reeling at a daily stream of troubling revelations about President Trump, the prospect that Vice President Mike Pence would assume power should Mr. Trump leave office is a remote possibility.

But for Democrats uttering the first whispers of impeachment talk in the halls of the Capitol, the president’s unobtrusive and strait-laced understudy has emerged in recent days as a prime target — an heir apparent to an increasingly embattled president whom they are eager to tarnish.

“It’s time to talk about Mike Pence,” Emily Aden, the rapid response director of American Bridge, a liberal political group, said in a memo circulated on Thursday. “Pence is just as complicit in this scandal as every other Republican in Washington, and despite his best efforts to fly under the radar, he should expect the country to hold him accountable.”

Once again this week, Mr. Pence emerged as a crucial player — either as a willful participant or an uninformed bystander — in the allegations that followed the revelation that Michael T. Flynn, Mr. Trump’s first national security adviser, had informed the White House weeks before the inauguration that he was under federal investigation for secretly working as a paid lobbyist for Turkey during the campaign.

Mr. Pence was leading the transition at the time, although his advisers said on Thursday that he stood by a comment he made in March, that he had only then learned of Mr. Flynn’s ties to Turkey.

It was the third time Mr. Pence had been forced to answer for public statements he made on behalf of the Trump administration that later turned out to be false. He asserted in a television interview in January that Mr. Flynn had not discussed sanctions with the Russian ambassador, only to learn later that Mr. Flynn had misrepresented that conversation, a breach that Mr. Trump cited in firing Mr. Flynn. It later emerged that Mr. Trump and senior White House officials had been well aware of Mr. Flynn’s dissembling for weeks before Mr. Pence learned of it.

Last week, Mr. Pence dutifully repeated the White House’s explanation for why Mr. Trump had abruptly fired James B. Comey, the F.B.I. director, telling a gaggle of reporters in the Capitol that the move had been based on a recommendation by the deputy attorney general, Rod J. Rosenstein, and centered on Mr. Comey’s mishandling of the investigation into Hillary Clinton’s use of a private email server. But the next day, the president said he had always intended to fire Mr. Comey, and requested Mr. Rosenstein’s input only after the decision had been made.

Despite the inconsistencies, some conservative Trump critics have begun openly musing about the prospect that Mr. Pence will succeed him.

“Republicans who are reflexively defending the self-inflicted wounds of this president have no need for him with Mike Pence in the wings,” Erick Erickson wrote on Wednesday on his conservative blog, The Resurgent.

Mr. Pence takes umbrage at the very mention of the idea, according to people who have heard him speak about it. And Republican lawmakers are eager to tamp down on any discussion of it, particularly after Mr. Rosenstein on Wednesday named a special counsel to lead the investigation into whether Mr. Trump’s campaign colluded with Moscow to influence the election.

“I’m not even going to give credence to that,” the House speaker, Paul D. Ryan of Wisconsin, said on Thursday, when asked by a reporter about colleagues who are privately contemplating a Pence presidency. “I’m not even — there’s not even a point making a comment on that.”

Still, the vice president may have fed speculation when he filed paperwork this week for a political action committee, an unusual move for a vice president.

Through it all, Mr. Pence has played the part of loyal soldier and messenger, gamely going about a workaday schedule that includes delivering Mr. Trump’s greetings to activists, lobbyists and lawmakers. He projects a “this is fine” outlook on the administration’s agenda of overhauling health care and the tax system.

“Whatever Washington, D.C., may be focused on at any given time, rest assured, President Donald Trump will never stop fighting for the issues that matter most to the American people — good jobs, safe streets and a boundless American future,” Mr. Pence told executives and lobbyists on Thursday at an investment summit meeting at the U.S. Chamber of Commerce.

A block away at the White House, Mr. Trump took a far different tack, telling a group of news anchors that the appointment of a special counsel to investigate his campaign’s possible collusion with Russia “hurts our country terribly.” Not long after that, with Mr. Pence in the front row at an East Room news conference, Mr. Trump declared that the inquiry was “a witch hunt” and that there was “no collusion” between him and Russia — though he quickly added, “I can always speak for myself.”

Mr. Pence rarely does so these days. His staff members say he is embarking on a busy schedule of speeches and travel to promote Mr. Trump’s legislative agenda, including health care, a tax overhaul and the president’s forthcoming budget, which will be unveiled while Mr. Trump is abroad for his first foreign trip. Starting on Thursday, Mr. Pence had seven speeches scheduled over four days, an aide said.

It is the steadying, practical role that Republicans had hoped Mr. Pence would play for a president who has no political or legislative experience and is openly disdainful of the details of policy. But with Mr. Trump’s White House increasingly embroiled in turmoil, and Mr. Pence repeatedly finding himself drawn into the storm, it is one he may not be positioned to play effectively.

“The idea for Republicans was that Pence is over there, Pence can be our liaison, Pence can help advance the agenda — a lot of chips were piled up on Pence as being the guarantor of the agenda and the guarantor of stability,” said David Axelrod, who was a top adviser to President Barack Obama. “It’s certainly undercut by these interludes of, ‘No one ever told me. I never knew that.’”

“It’s hard,” he said, “to look like the prime minister and the man in the dark at the same time.”